Theorem ssdifsym 3863

Description: Symmetric class differences for subclasses. (Contributed by AV, 3-Jan-2022.)

Assertion

Ref	Expression
ssdifsym	⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐵 = (𝑉 ∖ 𝐴) ↔ 𝐴 = (𝑉 ∖ 𝐵)))

Proof of Theorem ssdifsym

Step	Hyp	Ref	Expression
1		ssdifim 3862	. . . 4 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 = (𝑉 ∖ 𝐴)) → 𝐴 = (𝑉 ∖ 𝐵))
2	1	ex 450	. . 3 ⊢ (𝐴 ⊆ 𝑉 → (𝐵 = (𝑉 ∖ 𝐴) → 𝐴 = (𝑉 ∖ 𝐵)))
3	2	adantr 481	. 2 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐵 = (𝑉 ∖ 𝐴) → 𝐴 = (𝑉 ∖ 𝐵)))
4		ssdifim 3862	. . . 4 ⊢ ((𝐵 ⊆ 𝑉 ∧ 𝐴 = (𝑉 ∖ 𝐵)) → 𝐵 = (𝑉 ∖ 𝐴))
5	4	ex 450	. . 3 ⊢ (𝐵 ⊆ 𝑉 → (𝐴 = (𝑉 ∖ 𝐵) → 𝐵 = (𝑉 ∖ 𝐴)))
6	5	adantl 482	. 2 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐴 = (𝑉 ∖ 𝐵) → 𝐵 = (𝑉 ∖ 𝐴)))
7	3, 6	impbid 202	1 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐵 = (𝑉 ∖ 𝐴) ↔ 𝐴 = (𝑉 ∖ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∖ cdif 3571   ⊆ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by: (None)